Calculus II · Unit 4B · lesson

Differentiating and Integrating Power Series

Concept

Learning objectives

differentiate and integrate a power series term by term inside its radius of convergence.

Differentiating and Integrating Power Series

Explanation

Power series behave like polynomials inside their radius

Within the open interval of convergence, a power series may be differentiated or integrated term by term. The resulting series has the same radius of convergence, although endpoint behavior can change. This theorem turns one known series into a library of new identities.

Termwise operations are powerful because differentiation and integration act simply on monomials. They are also the first place where an infinite limiting process is exchanged with another operation. The radius condition supplies the uniform control needed for that exchange on smaller closed intervals, a fact explored more carefully in analysis.

Bridge

Inside the radius, a power series behaves like a polynomial

Within its radius of convergence, a power series may be differentiated or integrated term by term. Differentiation multiplies coefficients by the exponent and shifts powers downward; integration divides by the new exponent and shifts upward.

The radius remains the same, but endpoint behavior may change. Termwise operations are justified because convergence is uniformly controlled on every smaller closed interval strictly inside the radius. This prevents infinitely many small termwise errors from behaving badly when combined.

**Differentiation and integration shift coefficients and powers**
A row of terms maps downward under differentiation and upward under integration, with exponent multipliers or divisors displayed.
Differentiation and integration shift coefficients and powers. Coefficient-and-exponent shifts under differentiation and integration.

Proof idea

Uniform control keeps termwise operations honest

On $|x-a|\le r<R$ , the power-series terms are dominated by a convergent numerical series. This uniform bound allows limits, sums, derivatives, and integrals to interact safely. The full theorem belongs to analysis, but the key is control shared by every point in the smaller interval.

Concept

Termwise calculus

If $\sum c_n(x-a)^n$ has radius $R$ , then for $|x-a|<R$ ,

\frac{d}{dx}\sum c_n(x-a)^n=\sum_{n=1}^{\infty}nc_n(x-a)^{n-1},

and integration is also termwise.

Guided walkthrough

Derive a logarithm series

Start from

\frac1{1+x}=1-x+x^2-x^3+\cdots,\qquad |x|<1.

Integrate from $0$ to $x$ :

\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots.

Worked example

Derive the arctangent series from geometry's favorite series

From

\frac1{1+t^2}=1-t^2+t^4-t^6+\cdots\qquad(|t|<1),

integrate from $0$ to $x$ :

\arctan x=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\cdots.

The radius remains $1$ . Endpoint behavior must then be checked separately; at $x=1$ , the alternating series converges to $\pi/4$ .

Optional advanced note

Why this needs a theorem

Pointwise convergence alone does not justify interchanging a limit with differentiation. Power series possess stronger local control: on every closed interval strictly inside the radius, they converge uniformly. Uniform convergence is the analysis concept that keeps infinitely many small errors under control at once.

Common mistake

The radius stays, but endpoints may change

Do not copy the original interval of convergence mechanically. After differentiating or integrating, retest any finite endpoints.

Interactive checku4b-differentiating_and_integrating_power_series-01

Differentiate $\sum_{n=0}^{\infty}x^n$ term by term.

Your work stays on this device. No account or AI grader is used.

Show hint

The constant term vanishes and $d(x^n)/dx=nx^{n-1}$ .

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Exercise

Integrate $\sum x^n$ from $0$ to $x$ .

Exercise

Differentiate the series for $\ln(1+x)$ .

Exercise

Explain why radius stays the same while endpoints may change.

Exercise

Use termwise differentiation to find a series for $1/(1-x)^3$ .

After the explanation

Use the section idea

Reading lens

Treat algebra, differentiation, and integration as coefficient transformations with a preserved or rechecked interval.

Mental model

Inside a power series' convergence interval it behaves like a polynomial limit, so familiar operations become structured index shifts.

Decision

Write the source identity and validity interval before transforming coefficients or powers.

Common trap

Losing an index, constant of integration, or endpoint condition during a formal manipulation.

Check yourself

Can you reverse the operation and recover the source series?

Source & rights

Original instruction with traceable references.

BetterGrades-original; no direct adaptation declared in the verified handoff.

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.